Vanishing moments for scaling vectors

One advantage of scaling vectors over a single scaling function is the compatibility of symmetry and orthogonality. This paper investigates the relationship between symmetry, vanishing moments, orthogonality, and support length for a scaling vector Φ. Some general results on scaling vectors and vanishing moments are developed, as well as some necessary conditions for the symbol entries of a scaling vector with both symmetry and orthogonality. If orthogonal scaling vector Φ has some kind of symmetry and a given number of vanishing moments, we can characterize the type of symmetry for Φ, give some information about the form of the symbol P(z), and place some bounds on the support of each φi. We then construct an L2(R) orthogonal, symmetric scaling vector with one vanishingmoment having minimal support.